I struggle to solve the following minimization model with a nonlinear objective function and linear constraints. The objective function is the summation of two fraction terms with a quadratic numerator and linear denominator:
$min Z(x_1,y_1,x_2,y_2)=\frac{{\beta }_c{x_1}^2+{\beta }_mx_1y_1+{\beta }_n{y_1}^2}{2h_1(h_1-r_cx_1-r_ny_1)}+\frac{{\beta }_c{x_2}^2+{\beta }_mx_2y_2+{\beta }_n{y_2}^2}{2h_2(h_2-r_cx_2-r_ny_2)}$
s.t.
$x_1+x_2=1$
$y_1+y_2=1$
$0\le x_1,y_1,x_2,y_2\le 1$
All parameters are non-negative. Even the denominators are positive and nonzero. We can replace the equality constraints in the objective function and simplify the model as follows:
$min Z(x_1,y_1)=\frac{{\beta }_c{x_1}^2+{\beta }_mx_1y_1+{\beta }_n{y_1}^2}{2h_1(h_1-r_cx_1-r_ny_1)}+\frac{{\beta }_c{(1-x_1)}^2+{\beta }_m(1-x_1)(1-y_1)+{\beta }_n{(1-y_1)}^2}{2h_2(h_2-r_c(1-x_1)-r_n(1-y_1))}$
s.t.
$0\le x_1\le 1$
$0\le y_1\le 1$
But I wonder how this model with two variables can be analytically solved. The two terms in the objective function are symmetric and perhaps in the optimal solution, one of them takes a very low and the other takes a very high value to be most unbalanced.